Probably.
The United States does not have direct election for president. A Candidate is merely running for the right to appoint electors to the electoral college. The first direct vote for president occurs when the electoral college meets.
There is no Constitutional requirement that someone running for a slate of electors must be "eligible". He can direct his electors to vote for whoever he chooses.
Further the determination of whether or not Eldridge Cleaver should be determined to be ineligible is one for the voters, the electoral college and congress to make.
The states choose to put nominee names on the ballot. They don't have to do this. But if they choose to put the name of a living breating human being on the ballot, I submit that the states have the power to deny that to whoever the hell they want, for any reason at all. The states all play this hot potatoe (Hi Dan Quayle) game of requring a Certification of Qualification, but the states make no substantive test and require no evidence.
This facilitates the perpetration of fraud by the party or the candidate, whoever submits the certificate of qualification.